Beamforming Using Exact Evaluation of Leakage and Ergodic Capacity of MU-MIMO System

Closed-form evaluation of key performance indicators (KPIs) of telecommunication networks help perform mathematical analysis under several network configurations. This paper deals with a recent mathematical approach of indefinite quadratic forms to propose simple albeit exact closed-form expressions of the expectation of two significant logarithmic functions. These functions formulate KPIs which include the ergodic capacity and leakage rate of multi-user multiple-input multiple-output (MU-MIMO) systems in Rayleigh fading channels. Our closed-form expressions are generic in nature and they characterize several network configurations under statistical channel state information availability. As a demonstrative example of the proposed characterization, the derived expressions are used in the statistical transmit beamformer design in a broadcast MU-MIMO system to portray promising diversity gains using standalone or joint maximization techniques of the ergodic capacity and leakage rate. The results presented are validated by Monte Carlo simulations.

Siegel theta series for indefinite quadratic forms

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

This paper presents a rectangular branch-and-reduction algorithm for globally solving indefinite quadratic programming problem (IQPP), which has a wide application in engineering design and optimization. In this algorithm, first of all, we convert the IQPP into an equivalent bilinear optimization problem (EBOP). Next, a novel linearizing technique is presented for deriving the linear relaxation programs problem (LRPP) of the EBOP, which can be used to obtain the lower bound of the global optimal value to the EBOP. To obtain a global optimal solution of the EBOP, the main computational task of the proposed algorithm involves the solutions of a sequence of LRPP. Moreover, the global convergent property of the algorithm is proved, and numerical experiments demonstrate the higher computational performance of the algorithm.

Indefinite zeta functions

We define generalised zeta functions associated with indefinite quadratic forms of signature $$(g-1,1)$$ ( g - 1 , 1 ) —and more generally, to complex symmetric matrices whose imaginary part has signature $$(g-1,1)$$ ( g - 1 , 1 ) —and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at $$s=0$$ s = 0 are predicted to be logarithms of algebraic units by the Stark conjectures.

On approximating the distribution of indefinite quadratic expressions in singular normal vectors

General representations of quadratic forms and quadratic expressions in singular normal vectors are given in terms of the difference of two positive definite quadratic forms and an independentlydistributed linear combination of normal random variables. Up to now, only special cases have been treated in the statistical literature. The densities of the quadratic forms are then approximated with gamma and generalized gamma density functions. A moment-based technique whereby the initial approximations are adjusted by means of polynomials is presented. Closed form and integral formulae are provided for the approximate density functions of the quadratic forms and quadratic expressions. A detailed step-by-step algorithm for implementing the proposed density approximation technique is also provided. Two numerical examples illustrate the methodology.

AN INDEFINITE QUADRATIC PROGRAMMING PROBLEMS

In this paper, we give in section (1) compact description of the algorithm for solving general quadratic programming problems (that is, obtaining a local minimum of a quadratic function subject to inequality constraints) is presented. In section (2), we give practical application of the algorithm, we also discuss the computation work and performing by the algorithm and try to achieve efficiency and stability as possible as we can. In section (3), we show how to update the QR-factors of A1 (K), when the tableau is complementary ,we give updating to the LDLT-Factors of (K ) A G . In section (4) we are not going to describe a fully detailed method of obtaini

Trust Your Data or Not—StQP Remains StQP: Community Detection via Robust Standard Quadratic Optimization

We consider the robust standard quadratic optimization problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is optimized over the standard simplex. Following most approaches, we model the uncertainty sets by balls, polyhedra, or spectrahedra, more generally, by ellipsoids or order intervals intersected with subcones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of the aforementioned uncertainty sets and present conditions under which the RStQP reduces to the standard quadratic optimization problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.

Symmetry of topographs of Markoff forms

The Markoff spectrum is defined as the set of normalized values of arithmetic minima of indefinite quadratic forms. In the theory of the Markoff spectrum we observe various kinds of symmetry. Each of Conway’s topographs of quadratic forms which give values in the discrete part of the Markoff spectrum has a special infinite path consisting of edges. It has symmetry with respect to a translation along the path and countable central symmetries by which the path is invariant. We prove that these properties are obtained from the fact that the path is a discretization of a geodesic in the upper half-plane which corresponds to a value of the discrete part of the Markoff spectrum and projects to a simple closed geodesic on the once punctured torus with the highest degree of symmetry.